Base field \(\Q(\sqrt{19}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 19 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-19, 0, 1]))
gp: K = nfinit(Polrev([-19, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-19, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([-1,1]),K([1,1]),K([-351601,80668]),K([108562673,-24905977])])
gp: E = ellinit([Polrev([1,1]),Polrev([-1,1]),Polrev([1,1]),Polrev([-351601,80668]),Polrev([108562673,-24905977])], K);
magma: E := EllipticCurve([K![1,1],K![-1,1],K![1,1],K![-351601,80668],K![108562673,-24905977]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-18a+78)\) | = | \((-3a+13)^{3}\cdot(-a-4)\cdot(-a+4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
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Conductor norm: | \( 72 \) | = | \(2^{3}\cdot3\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
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Discriminant: | \((1296)\) | = | \((-3a+13)^{8}\cdot(-a-4)^{4}\cdot(-a+4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
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Discriminant norm: | \( 1679616 \) | = | \(2^{8}\cdot3^{4}\cdot3^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
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j-invariant: | \( \frac{1556068}{81} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
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Endomorphism ring: | \(\Z\) | ||
Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
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Sato-Tate group: | $\mathrm{SU}(2)$ |
Mordell-Weil group
Rank: | \(1\) | |
Generator | $\left(-255 a + 1108 : 10799 a - 47065 : 1\right)$ | |
Height | \(0.78129754193096045252450720825849982186\) | |
Torsion structure: | \(\Z/2\Z\oplus\Z/4\Z\) | |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
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Torsion generators: | $\left(-46 a + 197 : -76 a + 338 : 1\right)$ | $\left(-7 a + 27 : -1536 a + 6702 : 1\right)$ |
sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
|
BSD invariants
Analytic rank: | \( 1 \) | ||
sage: E.rank()
magma: Rank(E);
|
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Mordell-Weil rank: | \(1\) | ||
Regulator: | \( 0.78129754193096045252450720825849982186 \) | ||
Period: | \( 22.734034070006347604148590511773734758 \) | ||
Tamagawa product: | \( 64 \) = \(2^{2}\cdot2^{2}\cdot2^{2}\) | ||
Torsion order: | \(8\) | ||
Leading coefficient: | \( 4.0748925742799633517250737423966899635 \) | ||
Analytic order of Ш: | \( 1 \) (rounded) |
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-3a+13)\) | \(2\) | \(4\) | \(I_{1}^{*}\) | Additive | \(-1\) | \(3\) | \(8\) | \(0\) |
\((-a-4)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
\((-a+4)\) | \(3\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
72.1-j
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:
Base field | Curve |
---|---|
\(\Q\) | 48.a3 |
\(\Q\) | 8664.j3 |